(0) Obligation:
Clauses:
f(X) :- g(s(s(s(X)))).
f(s(X)) :- f(X).
g(s(s(s(s(X))))) :- f(X).
Query: f(g)
(1) PrologToPrologProblemTransformerProof (SOUND transformation)
Built Prolog problem from termination graph ICLP10.
(2) Obligation:
Clauses:
fA(s(T8)) :- fA(T8).
fA(s(T12)) :- fA(T12).
Query: fA(g)
(3) PrologToPiTRSProof (SOUND transformation)
We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
fA_in: (b)
Transforming
Prolog into the following
Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:
fA_in_g(s(T8)) → U1_g(T8, fA_in_g(T8))
fA_in_g(s(T12)) → U2_g(T12, fA_in_g(T12))
U2_g(T12, fA_out_g(T12)) → fA_out_g(s(T12))
U1_g(T8, fA_out_g(T8)) → fA_out_g(s(T8))
The argument filtering Pi contains the following mapping:
fA_in_g(
x1) =
fA_in_g(
x1)
s(
x1) =
s(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
U2_g(
x1,
x2) =
U2_g(
x2)
fA_out_g(
x1) =
fA_out_g
Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
(4) Obligation:
Pi-finite rewrite system:
The TRS R consists of the following rules:
fA_in_g(s(T8)) → U1_g(T8, fA_in_g(T8))
fA_in_g(s(T12)) → U2_g(T12, fA_in_g(T12))
U2_g(T12, fA_out_g(T12)) → fA_out_g(s(T12))
U1_g(T8, fA_out_g(T8)) → fA_out_g(s(T8))
The argument filtering Pi contains the following mapping:
fA_in_g(
x1) =
fA_in_g(
x1)
s(
x1) =
s(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
U2_g(
x1,
x2) =
U2_g(
x2)
fA_out_g(
x1) =
fA_out_g
(5) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:
FA_IN_G(s(T8)) → U1_G(T8, fA_in_g(T8))
FA_IN_G(s(T8)) → FA_IN_G(T8)
FA_IN_G(s(T12)) → U2_G(T12, fA_in_g(T12))
The TRS R consists of the following rules:
fA_in_g(s(T8)) → U1_g(T8, fA_in_g(T8))
fA_in_g(s(T12)) → U2_g(T12, fA_in_g(T12))
U2_g(T12, fA_out_g(T12)) → fA_out_g(s(T12))
U1_g(T8, fA_out_g(T8)) → fA_out_g(s(T8))
The argument filtering Pi contains the following mapping:
fA_in_g(
x1) =
fA_in_g(
x1)
s(
x1) =
s(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
U2_g(
x1,
x2) =
U2_g(
x2)
fA_out_g(
x1) =
fA_out_g
FA_IN_G(
x1) =
FA_IN_G(
x1)
U1_G(
x1,
x2) =
U1_G(
x2)
U2_G(
x1,
x2) =
U2_G(
x2)
We have to consider all (P,R,Pi)-chains
(6) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FA_IN_G(s(T8)) → U1_G(T8, fA_in_g(T8))
FA_IN_G(s(T8)) → FA_IN_G(T8)
FA_IN_G(s(T12)) → U2_G(T12, fA_in_g(T12))
The TRS R consists of the following rules:
fA_in_g(s(T8)) → U1_g(T8, fA_in_g(T8))
fA_in_g(s(T12)) → U2_g(T12, fA_in_g(T12))
U2_g(T12, fA_out_g(T12)) → fA_out_g(s(T12))
U1_g(T8, fA_out_g(T8)) → fA_out_g(s(T8))
The argument filtering Pi contains the following mapping:
fA_in_g(
x1) =
fA_in_g(
x1)
s(
x1) =
s(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
U2_g(
x1,
x2) =
U2_g(
x2)
fA_out_g(
x1) =
fA_out_g
FA_IN_G(
x1) =
FA_IN_G(
x1)
U1_G(
x1,
x2) =
U1_G(
x2)
U2_G(
x1,
x2) =
U2_G(
x2)
We have to consider all (P,R,Pi)-chains
(7) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 2 less nodes.
(8) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FA_IN_G(s(T8)) → FA_IN_G(T8)
The TRS R consists of the following rules:
fA_in_g(s(T8)) → U1_g(T8, fA_in_g(T8))
fA_in_g(s(T12)) → U2_g(T12, fA_in_g(T12))
U2_g(T12, fA_out_g(T12)) → fA_out_g(s(T12))
U1_g(T8, fA_out_g(T8)) → fA_out_g(s(T8))
The argument filtering Pi contains the following mapping:
fA_in_g(
x1) =
fA_in_g(
x1)
s(
x1) =
s(
x1)
U1_g(
x1,
x2) =
U1_g(
x2)
U2_g(
x1,
x2) =
U2_g(
x2)
fA_out_g(
x1) =
fA_out_g
FA_IN_G(
x1) =
FA_IN_G(
x1)
We have to consider all (P,R,Pi)-chains
(9) UsableRulesProof (EQUIVALENT transformation)
For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
(10) Obligation:
Pi DP problem:
The TRS P consists of the following rules:
FA_IN_G(s(T8)) → FA_IN_G(T8)
R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
(11) PiDPToQDPProof (EQUIVALENT transformation)
Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
(12) Obligation:
Q DP problem:
The TRS P consists of the following rules:
FA_IN_G(s(T8)) → FA_IN_G(T8)
R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
(13) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- FA_IN_G(s(T8)) → FA_IN_G(T8)
The graph contains the following edges 1 > 1
(14) YES